2013年最新美国数学竞赛试题及答案(AMC2013A)

A

1

Square

has side length

. Point

is on

, and the area of

is

. What is

?

2

A softball team played ten games, scoring , and runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

3

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

4 What is the value of

5

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid , Dorothy paid , and Sammy paid . In order to share the costs equally, Tom ? gave Sammy dollars, and Dorothy gave Sammy dollars. What is

6

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on of her three-point shots and of her two-point shots. Shenille attempted shots. How many points did she score?

7

The sequence sum of the previous two. That is, . What is ?

has the property that every term beginning with the third is the Suppose that and

8

Given that

and

are distinct nonzero real numbers such that

, what is

?

9

In , respectively, such that of parallelogram ?

and and

. Points are parallel to

and are on sides , , and , and , respectively. What is the perimeter

(9th)

（11

th

）

10

Let

be the set of positive integers with and

for which

has the repeating decimal representation ? are on

different digits. What is the sum of the elements of . Points and are on and points ? and

11 Triangle and

is equilateral with and

such that both

are parallel to

. Furthermore, triangle

and trapezoids

all have the same perimeter. What is

12 he angles in a particular triangle are in arithmetic progression, and the side lengths are sum of the possible values of ? 13 Let points equals where , and

. The

are positive integers. What is

and

. Quadrilateral

is cut into

equal area pieces by a line passing through fractions are in lowest terms. What is 14 The sequence , , , , is an arithmetic progression. What is ?

. This line intersects ?

at point

, where these

15 Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done? 16 , , are three piles of rocks. The mean weight of the rocks in is is and pounds, the mean weight and is and ? is pounds. What is the

of the rocks in

pounds, the mean weight of the rocks in the combined piles

pounds, and the mean weight of the rocks in the combined piles

greatest possible integer value for the mean in pounds of the rocks in the combined piles 17 A group of

pirates agree to divide a treasure chest of gold coins among themselves as follows. The of the coins that remain in the chest. The number of coins initially

pirate to take a share takes

in the chest is the smallest number for which this arrangement will allow each pirate to receive a

positive whole number of coins. How many coins doe the

pirate receive?

18 Six spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?

19 In and 20 Let

, . Moreover be the set

, and and

. A circle with center

and radius ?

intersects

at points

have integer lengths. What is . For , define

to mean that either have the property that ,

or

. How many ordered triples , and ? 21 Consider following intervals contains

of elements of

. Which of the ?

22 A palindrome is a nonnegatvie integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is the probability that is also a palindrome? is chosen uniformly at random. What

23

is a square of side length bounded by is rotated area is , where ,

. Point is on such that . The square region counterclockwise with center , sweeping out a region whose are positive integers and . What is ?

, and

24 Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?